The Annals of Applied Probability

Limit theory for point processes in manifolds

Mathew D. Penrose and J. E. Yukich

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Let $Y_{i}$, $i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal{M}\subset\mathbb{R}^{d}$ and consider sums $\sum_{i=1}^{n}\xi(n^{1/m}Y_{i},\{n^{1/m}Y_{j}\}_{j=1}^{n})$, where $\xi$ is a real valued function defined on pairs $(y,\mathcal{Y} )$, with $y\in\mathbb{R}^{d}$ and $\mathcal{Y}\subset\mathbb{R}^{d}$ locally finite. Subject to $\xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $\xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $\mathcal{M} $. We apply the general results to establish the limit theory of dimension and volume content estimators, Rényi and Shannon entropy estimators and clique counts in the Vietoris–Rips complex on $\{Y_{i}\}_{i=1}^{n}$.

Article information

Ann. Appl. Probab., Volume 23, Number 6 (2013), 2161-2211.

First available in Project Euclid: 22 October 2013

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Manifolds dimension estimators entropy estimators Vietoris–Rips complex clique counts


Penrose, Mathew D.; Yukich, J. E. Limit theory for point processes in manifolds. Ann. Appl. Probab. 23 (2013), no. 6, 2161--2211. doi:10.1214/12-AAP897.

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